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重点关注论文（完整分析）
1. 打破超材料均匀化中的尺度分离：界面惯性增强的松弛微态模型
Breaking Scale Separation in Metamaterials’ homogenization: Interface-Inertia-Enhanced Relaxed Micromorphic Model
作者: Angela Madeo, Leonardo Perez, Mohammad Sarhil
类别: math.NA
发布日期: 2026-07-29
arXiv ID: 2607.27385v1
摘要翻译: 均匀化连续体模型被广泛用于描述力学超材料中的波传播和带隙行为，而无需显式解析其微结构。然而，这些模型的有效性通常依赖于经典的尺度分离假设，即传播扰动的波长远大于单胞的特征尺寸。在有限尺寸超材料样本中及较高频率下，这一假设逐渐被打破，动态响应开始强烈受到微结构在外边界处截断方式的影响。在这项工作中，我们在力学超材料的均匀化描述中引入了一个全新的概念：宏观界面的惯性贡献。我们表明，同一晶格的不同截断会产生具有不同质量分布的边界，这导致有限尺寸试样的动态响应出现可测量的差异。为了在均匀化框架中捕捉这一复杂机制，我们扩展了松弛微态模型，在所考虑物体的边界上引入了动能表面能。这在边界条件中产生了一个额外的惯性项，可被视为微结构截断所产生的界面惯性的均匀化对应物。因此，均匀化模型现在能够区分具有相同体性质但仅界面构型不同的有限尺寸试样。所提出的公式保留了松弛微态模型的变分结构，同时使连续体能够重现全解析模拟中观察到的依赖边界的响应，特别是在频率范围内……. 完整摘要请参见PDF。
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链接: http://arxiv.org/abs/2607.27385v1
1. 研究对象和背景
机械超材料通过微结构几何与弹性波的相互作用表现出非凡的动态特性，如带隙（频率范围内波无法传播）和定向波导。这些现象通常由局部共振或布拉格散射引起，吸引了均匀化理论的大量研究。传统的连续介质模型（如柯西弹性）的色散关系是线性的，无法描述色散行为；因此，发展了多种广义连续体理论。其中，松弛微态模型（relaxed micromorphic model）因其能准确再现周期微结构介质的色散曲线和带隙行为而得到广泛应用。该模型引入宏观位移场 $u\in\mathbb{R}^3$ 和非对称微畸变张量 $P\in\mathbb{R}^{3\times 3}$ 作为基本未知量，其 Lagrangian 密度为 $\mathcal{L}_{\text{rrm}} = K(\dot u, \nabla\dot u, \dot P) - W(\nabla u, P)$，其中动能和应变能分别为
$$ \begin{aligned} K(\dot u, \nabla\dot u, \dot P) &= \frac12 \rho\langle\dot u,\dot u\rangle + \frac12\langle\mathbb J_m\operatorname{sym}\dot P,\operatorname{sym}\dot P\rangle + \frac12\langle\mathbb J_c\operatorname{skew}\dot P,\operatorname{skew}\dot P\rangle \\ &\quad + \frac12\langle\mathbb T_e\operatorname{sym}\nabla\dot u,\operatorname{sym}\nabla\dot u\rangle + \frac12\langle\mathbb T_c\operatorname{skew}\nabla\dot u,\operatorname{skew}\nabla\dot u\rangle, \end{aligned} $$$$ W(\nabla u, P) = \frac12\langle\mathbb C_e\operatorname{sym}(\nabla u-P),\operatorname{sym}(\nabla u-P)\rangle + \frac12\langle\mathbb C_c\operatorname{skew}(\nabla u-P),\operatorname{skew}(\nabla u-P)\rangle + \frac12\langle\mathbb C_{\text{micro}}\operatorname{sym} P,\operatorname{sym} P\rangle, $$这里 $\rho$ 是宏观表观密度，$\mathbb J_m,\mathbb J_c,\mathbb T_e,\mathbb T_c$ 是四阶微惯性张量，$\mathbb C_e,\mathbb C_c,\mathbb C_{\text{micro}}$ 是四阶弹性张量。通过适当标定，该模型能在无限或“等效无限”周期域中精确描述波传播和带隙，前提是波长 λ 远大于单胞尺寸 L，即经典的尺度分离假设成立。
然而，实际应用中的超材料构件常常是有限尺寸的，仅由少量单胞构成。当频率升高，波长缩小到与单胞尺寸可比时，尺度分离假设逐渐失效，结构的响应强烈依赖于外部边界如何截断微结构。例如，同一周期性无限晶格沿不同平面截断将产生具有不同质量分布和几何构型的边界（如α-截断、β-截断、γ-截断）。在全解析微结构仿真中，这些差异自然体现，但标准均匀化模型仅包含体能和体动能，无法区分此类界面差异。因此，传统均匀化模型预测的散射场、牵引分布等与精细仿真结果出现显著偏差，尤其在高频段。
本文旨在填补这一空白：首次在均匀化框架中引入宏观界面的惯性贡献。基本思想是在松弛微态模型的 Lagrangian 中附加一个定义于边界 ∂Ω 的动能面能量，其形式可能为 $\int_{\partial\Omega} \frac12 \langle\mathbb I^{\text{surface}}\dot u,\dot u\rangle\,dA$ 或类似含微畸变的表达式，这里 $\mathbb I^{\text{surface}}$ 是与截断方式相关的界面惯性张量。该面能量通过变分原理在边界条件中生成额外的惯性项，从而赋予连续体模型区分不同界面配置的能力，同时保持模型的变分一致性。这一概念从根本上打破了均匀化中长期依赖的尺度分离假设，为预测有限尺寸超材料构件的高频动力学响应开辟了新途径。
2. 主要定理或主要结果
本文的主要贡献是提出了一种界面惯性增强的松弛微态模型，并展示了其在捕捉截断依赖的动态响应方面的有效性。具体结论如下：
- 模型构建：在原有的缩减松弛微态模型（reduced relaxed micromorphic model）基础上，通过引入边界上的动能面能量，扩展了作用量泛函。新的作用量形如 $$\mathcal A^{\text{enhanced}} = \int_0^\tau\int_\Omega \mathcal L_{\text{rrm}}\,dx\,dt + \int_0^\tau\int_{\partial\Omega} \mathcal K^{\text{surface}}(\dot u,\dot P)\,dA\,dt,$$ 其中 $\mathcal K^{\text{surface}}$ 是依赖于界面惯性张量 $\mathbb I^{\text{surface}}$ 的二次型。通过变分得到修正的运动方程和自然边界条件，后者包含由表面惯性导致的附加惯性项（如 $\mathbb I^{\text{surface}}\ddot u$）。
- 数值验证：以“十字”形单胞构成的超材料块为算例，对比了α-截断和β-截断的全解析微结构仿真（有限元离散单胞细节）与均匀化模型预测。在基准散射问题中（平面波入射到嵌入均匀介质的有限超材料块），观察到：
- 当波长 $\lambda > 10.33L$（L为单胞尺寸）时，两种截断的位移场几乎无差异，尺度分离假设成立，传统模型可接受。
- 当 $\lambda \le 10.33L$ 时，α-截断与β-截断的散射场出现显著差别，且这些差别在 $\lambda$ 接近L时愈加突出。 引入界面惯性后，均匀化模型能够再现全解析仿真中观察到的截断依赖性差异，尤其在高频段，而传统模型（仅含体能量）无法区分不同截断。
- 通用性：界面惯性张量 $\mathbb I^{\text{surface}}$ 可由截断面的质量分布直接确定，与体材料参数无关，因此模型可推广至任意晶格截断形式。文中给出了α、β、γ三种截断对应的界面惯性张量示例。
- 变分一致性：所提模型严格保持了变分结构，从而确保了能量守恒等物理合理性，并可自然地与有限元等数值方法结合。
需要指出，以上结果中关于界面惯性张量的具体辨识方法及数值例中均匀化模型与全解析仿真的定量误差分析，在提供的PDF摘录中未完全展开，仅定性展示了差异的捕捉能力。
3. 研究方法、关键技术和核心工具
本文的核心创新在于将界面惯性作为固有部分引入均匀化连续体的变分框架中，而非作为外部加载或约束处理。技术路线可分为两步：
1. 界面动能面能量的采纳与变分推导
传统松弛微态模型的作用量仅包含体域Ω上的积分。为反映边界截断带来的局部惯性变化，作者直接在其Lagrangian中增添了一个定义在边界∂Ω上的动能面能量项。此项被假定为依赖于界面速度（$\dot u|_{\partial\Omega}$ 以及可能 $\dot P|_{\partial\Omega}$）的二次型，由界面惯性张量 $\mathbb I^{\text{surface}}$ 参数化。这是一个唯象但物理动机明确的做法：不同截断平面穿过单胞时留下不同的固体材料边界，从而具有不同的面密度，恰如薄板的惯性。
通过 Hamilton 原理（作用量驻值）对扩展作用量进行变分，得到了域内与经典模型相同的平衡方程
其中应力 $\sigma$ 与相对畸变 $(\nabla u - P)$ 及微畸变 $P$ 相关。关键在于自然边界条件变为
$$\sigma n = \mathbb I^{\text{surface}}\ddot u + \text{可能的其他惯性项},$$这里 $n$ 为边界外法向。该条件表明，边界上的牵引力不仅与内部应力平衡，还须克服界面自身的惯性，从而在动力学上区分不同截断。此项推导是标准变分法，但将表面惯性纳入 Lagrangian 系首次提出。
2. 界面惯性张量的确定与数值验证
为赋予 $\mathbb I^{\text{surface}}$ 具体的物理意义，作者考虑了单胞的截断方式。以“十字”形单胞为例，α、β、γ三种截断分别沿着不同平面切割，使边界上固体材料（铝）的分布不同（图1,3）。通过计算截断面上固体区域的面质量分布，可构造界面惯性张量。具体而言，若边界面上位置 $x$ 处的材料指示函数为 $\chi(x)$，则面密度张量为 $\int_{\text{surface}} \rho_{\text{solid}}\chi(x)\,ds$，并以某种方式转化为宏观惯性张量 $\mathbb I^{\text{surface}}$。PDF未详述此映射，但显然可利用均匀化理论中的面积分或平均方法。
为展示模型效果，论文进行了全解析微结构仿真（使用有限元精细离散所有单胞细节）与所提均匀化模型的对比。仿真设置了一个基准散射问题：5×5单胞的有限超材料块嵌入均匀基质中，受平面压缩波入射。通过比较沿两条检测线的位移剖面（图5-8），观察到传统均匀化模型（无界面惯性）无法预测α与β截断的差别，而增强模型能够再现这些差别，尤其当波长 $\lambda\le10.33L$ 时。验证中未给出误差阶或收敛速率，但定性确认了模型捕捉边界惯性效应的能力。
技术障碍与创新点
主要障碍在于，在尺度分离失效时，均匀化模型如何处理因截断导致的非均匀边界效应。传统做法是精细离散化微结构，但计算代价高昂。本文的创新点是直接提升连续体模型的复杂度——在变分层面嵌入表面惯性项，使宏观模型无需解析微结构即可捕获边界依赖的动态行为。关键引理或估计未以定理形式呈现，但实质是通过修正边界条件将微观截断信息编码进宏观参数 $\mathbb I^{\text{surface}}$。证明框架未涉及严格收敛性分析，而侧重于物理建模与数值例证。
4. 与之前工作的比较
本文在以下方面与既有均匀化及高阶连续体力学研究形成对比和推进：
1. 与传统均匀化方法（渐近均匀化、计算均匀化）的比较
传统均匀化依赖严格的尺度分离，仅在无限大周期域的前提下方能导出有效本构（如通过单元胞问题求解），且自然边界条件通常退化为柯西牵引条件。即便在有限尺寸问题中施加周期性边界条件，亦无法捕捉因截断造成的界面差异。本文放弃尺度分离假设，直接引入边界惯性项，使得均匀化模型能在波长与单胞尺寸可比时区分不同截断构型，拓宽了均匀化模型的适用频率范围。代价是需要额外标定界面惯性张量，此张量依赖于截断方式但独立于体性质，可通过简单的面积分预计算获得。
2. 与广义连续体模型（应变梯度、微态、微极等）中的表面效应对比
广义连续体理论中早有表面效应的研究，如 Mindlin 的应变梯度理论可包含表面应变能，Gurtin–Murdoch 的表面弹性模型引入表面本构关系。然而，这些表面能几乎全是应变能或自由能，反映的是界面处因原子排列不同而导致的额外弹性。本文所提的界面惯性是动能面能量，代表界面处因微结构截断而产生的额外质量效应，这是全新的物理机制。在动力学中，表面惯性通过修正边界条件直接影响波的反射、散射行为，与表面弹性作用在本质上是正交的。
3. 与微观界面力模型（如界面的内聚力或摩擦模型）的比较
作者在脚注中明确区分：许多工作考虑了微观层面（单元胞内部相界面）的界面力对宏观行为的影响，而本文考虑的是宏观构件外边界上的界面作用，它是由晶格截断产生的全局效应，是界面自身的一种均匀化属性。因此，本文的界面惯性是宏观本构的一部分，而非微观界面的直接升尺度。这种处理方式在均匀化文献中尚无先例。
4. 与松弛微态模型已有工作的对比
松弛微态模型已被Madeo等用于研究无限大和有限尺寸超材料，例如[19,21,25,32,43]等。这些工作主要集中在域内波传播和色散关系，边界条件多为经典牵引或固定位移。本文在同一模型框架下，通过添加表面动能，使边界条件获得惯性修正，从而能够处理尺度分离破坏时的界面效应。与[43]中观察到的界面牵引差异相呼应，本文提供了解决该差异的理论方案。此外，本文首次建立了界面惯性张量与截断方式的具体联系，使模型具备预测能力。
5. 适用范围与局限性
推进之处：模型将均匀化的适用频率扩展到尺度分离条件以下，即 λ≲10L 的区域，使有限尺寸超材料构件的高频响应可高效计算。局限性：界面惯性张量依赖于截断方式，需对每种截断预先计算，但对于同一材料不同截断的构件，可重复使用；文中尚未给出界面惯性张量的严格解析公式，可能依赖于数值拟合；仅限于线弹性、小变形，未涉及非线性或耗散。从提供内容中，作者未显式比较与具体的前人文献的定量差异，但清晰的物理动机和数值证据支持了其优越性。
了解领域论文（摘要翻译）
1. 保结构积分的统一离散梯度-SAV框架
A Unified Discrete Gradient-SAV Framework for Structure-Preserving Integration
作者: Elena Celledoni, David Martín de Diego, Brynjulf Owren, Miguel Vaquero
类别: math.NA, math-ph
发布日期: 2026-07-30
arXiv ID: 2607.27795v1
链接: http://arxiv.org/abs/2607.27795v1
摘要翻译
中文标题: 保结构积分的统一离散梯度-SAV框架
摘要翻译: 我们提出了一个将离散梯度(DG)方法与标量辅助变量(SAV)方法相结合的框架，用于为耗散系统和保守系统构造保结构积分器。关键观察是，SAV二次化将动力学提升到一个扩展的状态空间，在该空间上修正能量具有精确的离散梯度恒等式。这一观点产生了三种具有不同精度和计算代价的积分器：一阶半隐式向前Euler格式，二阶自伴中点格式，以及具有降低的隐式程度的二阶预测格式。该构造可扩展到几乎Poisson系统，并在可实施的离散条件下保持选定的Casimir不变量。数值实验涵盖了Allen-Cahn方程，Ohta-Kawasaki型非局部梯度流，双阱Hamilton振子，以及具有非线性三次Casimir的Poisson系统。
2. 一种用于多尺度线性动理学输运方程的渐近保持动力学低秩半拉格朗日方法
An Asymptotic-Preserving Dynamical Low-Rank Semi-Lagrangian Method for Multiscale Linear Kinetic Transport Equations
作者: Shun Li, Yan Jiang, Mengping Zhang, Tao Xiong
类别: math.NA
发布日期: 2026-07-30
arXiv ID: 2607.27736v1
链接: http://arxiv.org/abs/2607.27736v1
摘要翻译
中文标题: 一种用于多尺度线性动理学输运方程的渐近保持动力学低秩半拉格朗日方法
摘要翻译: 本文提出了一种用于多尺度线性动理学输运方程的渐近保持（AP）动力学低秩半拉格朗日方法。该方法结合了半拉格朗日离散化的大时间步长能力与低秩表示所提供的存储和计算成本减少。所提出的格式将近似宏观密度更新与用于动理学分布的基更新 Galerkin 积分器耦合起来。为了在半拉格朗日通量评估中保持降低的复杂度，通量导数通过采样角度求积策略计算。我们建立了常系数情形下全求积低秩格式的无条件稳定性分析。通量导数中角度采样引起的误差被量化。得到的格式在扩散极限下被证明是渐近保持的。数值实验，包括高维测试例子，表明所提出的方法在动理学和扩散区间内是渐近保持的，在大时间步长下稳定，且计算高效。
3. 面向浅水溃坝问题的参数化物理信息神经网络与张量降阶模型的比较
Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem
作者: Anton Myshak, Md Rezwan Bin Mizan, Ilya Timofeyev
类别: math.NA, cs.LG, physics.flu-dyn
发布日期: 2026-07-29
arXiv ID: 2607.27433v1
链接: http://arxiv.org/abs/2607.27433v1
摘要翻译
中文标题: 面向浅水溃坝问题的参数化物理信息神经网络与张量降阶模型的比较
摘要翻译: 我们建立了两种参数化数据驱动的降阶模型：物理信息神经网络（PINN）和非侵入式张量降阶模型（TROM），并将这两种方法应用于参数化一维浅水溃坝问题。两种降阶模型均无需时间积分，而是学习从空间、时间和溃坝参数到物理状态的直接解映射。我们针对样本外参数值和外推参数值进行了详细比较。此外，我们证明引入激波感知的配点对于提高PINN模型的鲁棒性至关重要。
不相关论文（基本信息）
1. 面向迭代求解器的图神经网络多层预条件子
Graph Neural Multilevel Preconditioners for Iterative Solvers
作者: Zechen Zhang, Rui Peng Li, Yousef Saad
类别: math.NA, cs.LG
发布日期: 2026-07-30
arXiv ID: 2607.28456v1
链接: http://arxiv.org/abs/2607.28456v1
摘要: Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning. While classical methods such as algebraic multigrid (AMG) are highly scalable, their robustness can degrade on indefinite or nonsymmetric systems where heuristics originally developed for elliptic PDEs are less reliable. Recently, Graph Neural Networks (GNNs) have emerged as data-driven preconditioners; yet, the practical impact of imposing an AMG-style hierarchy remains underexplored for general sparse matrices. In this work, we propose a Graph Neural Multilevel Preconditioner (GMP) that adopts an AMG hierarchy as a structural prior and learns smoothing, restriction, and interpolation operators in a unified framework. Our method targets general sparse systems and is instantiated as a drop-in preconditioner for standard Krylov solvers. On a benchmark of over 800 sparse matrices, we compare against classical AMG, single-level ILUT, and state-of-the-art GNN preconditioners, and characterize the regimes where multilevel graph neural preconditioning improves convergence or, conversely, introduces overhead relative to strong single-level baselines. These results highlight both the promise and the limitations of enforcing AMG-style multilevel structure in learned preconditioners for large-scale scientific simulations.
2. 弹跳粒子与Zigzag采样器的窗口化稀疏化和查询复杂度
Windowed thinning and query complexity for the bouncy particle and Zigzag samplers
作者: Jianfeng Lu, Yinchen Luo
类别: math.NA, cs.LG, math.PR, math.ST
发布日期: 2026-07-30
arXiv ID: 2607.28413v1
链接: http://arxiv.org/abs/2607.28413v1
摘要: Let $μ(d x)\propto e^{-U(x)} d x$ on $\R^d$, where $U$ is $m$-strongly convex and $L$-smooth, and denote by $κ=L/m$ the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error $\varepsilon$, the expected query counts are $O(κ^{1/2}d\,(d\logκ+\log\frac1\varepsilon))$ gradient queries for the bouncy particle sampler and $O(κd^{1/4}(d\logκ+\log\frac1\varepsilon))$ full-gradient equivalents for Zigzag, where $d$ coordinate-partial queries count as one equivalent.
3. 非对称加权 Hermite 近似应用于 Vlasov-Poisson 系统的 Galerkin 方法
Galerkin method for asymmetrically-weighted Hermite approximations applied to the Vlasov-Poisson system
作者: Ruiyang Dai
类别: math.NA
发布日期: 2026-07-30
arXiv ID: 2607.28409v1
链接: http://arxiv.org/abs/2607.28409v1
摘要: We investigate a numerical method for the Vlasov-Poisson (VP) system utilizing asymmetrically-weighted (AW) Hermite bases in velocity space, which is an hyperbolic system. In particular, we concentrate on spectral methods in velocity. For the Hermite spectral form of the VP system, we analyze the resaon that the form with AW Hermite bases can be instable. To obtain L2 stability properties, we consider a Galerkin method intead of the classical Petrov-Galerkin method, which naturally ensures stability with respect to the L2 norm. We also present an equivalent form of the method that maintains a computational cost modest compared to that of the Petrov-Galerkin method. Finally, we present numerical simulations based on the proposed Hermite spectral method, showcasing its stability.
4. 一种用于双乘积节省下矩阵多项式求值方案稳定生成的 MATLAB 工具
A MATLAB Tool for the Stable Generation of Matrix Polynomial Evaluation Schemes with Two-Product Savings
作者: J. Ibáñez, J. Sastre, J. M. Alonso, E. Defez
类别: math.NA
发布日期: 2026-07-30
arXiv ID: 2607.28286v1
链接: http://arxiv.org/abs/2607.28286v1
摘要: Computing numerical approximations of matrix functions frequently relies on the efficient evaluation of high-degree matrix polynomials. Although computational bounds are historically governed by the Paterson–Stockmeyer (PS) method, recent theoretical developments have demonstrated the viability of evaluation schemes that eliminate two matrix products ($2M$). Existing literature documents stable instances of this $2M$ reduction only for isolated cases, such as specific degrees of Taylor approximations for the matrix exponential and the matrix logarithm. However, a generalized approach for arbitrary polynomials remains unestablished. To address this limitation, this work presents a software-driven procedure that extends these computational savings to polynomials of degrees $m \in \{18, 21, 24, 26, 27, 28\}$ and all $m \ge 30$, requiring primarily a non-zero leading coefficient. Since the underlying evaluation coefficients must be determined by solving systems of nonlinear equations (SNEs), selecting a numerically stable solution set is critical. We introduce an automated verification routine designed to filter and validate robust coefficient sets for floating-point execution. The primary contribution is a MATLAB implementation leveraging variable precision arithmetic to handle the underlying SNEs, verify stability, and project precision bounds. Numerical experiments involving various matrix functions verify that the developed implementation preserves or, in some instances, enhances the numerical accuracy of the classic PS method, while systematically achieving the theoretical reduction of $2M$.
5. R3MG-C：一种用于连续有限元离散的高阶代数-几何多层预处理子
R3MG-C: a high-order algebraic-geometric multilevel preconditioner for continuous finite element discretizations
作者: Davide Polverino, Marco Feder, Luca Heltai
类别: math.NA
发布日期: 2026-07-30
arXiv ID: 2607.28235v1
链接: http://arxiv.org/abs/2607.28235v1
摘要: Algebraic multigrid (AMG) methods are robust and efficient black-box preconditioners for linear systems arising from low-order discretizations of elliptic partial differential equations, but their performance often deteriorates for high-order methods. We introduce an algebraic-geometric multilevel preconditioner for continuous lagrangian finite element discretizations. The method automatically constructs prolongation operators and a Galerkin hierarchy using only the coordinates of finite element support points, without requiring a prescribed mesh hierarchy or domain decomposition. The support-point are recursively partitioned by an R-tree algorithm based on axis-aligned bounding boxes, producing a hierarchy of agglomerates. In contrast to classical AMG, which typically uses piecewise-constant aggregate coarse spaces, our method embeds local discontinuous polynomial spaces of degree $p'>0$, defined on the agglomerate boxes, through continuous nodal interpolation. This yields a high-order extension of the Nicolaides coarse space. A two-level analysis quantifies how the coarse polynomial degree offsets the effects of large agglomerates and high-order fine discretizations, under uniform box-regularity, interpolation-stability, and stable-decomposition assumptions. Numerical experiments in two and three dimensions show that the resulting V-cycle, used as a conjugate-gradient preconditioner, maintains stable iteration counts and remains effective in regimes where standard AMG deteriorates.
6. 分层网格上标记驱动水平集输运的自适应有限元方法
An Adaptive Finite Element Method for Marker-Driven Level-Set Transport on Hierarchical Meshes
作者: Eugenio Aulisa, Samuele Baldini, Giacomo Barbi, Andrea Chierici, Sandro Manservisi
类别: math.NA
发布日期: 2026-07-30
arXiv ID: 2607.28104v1
链接: http://arxiv.org/abs/2607.28104v1
摘要: This work presents a new adaptive finite-element framework for level-set transport, achieving high accuracy in kinematic interface transport problems relevant to interface-capturing methods for two-phase flows while reducing computational cost. This framework accommodates dynamic refinement and coarsening, and is compatible with standard finite-element data structures. The algorithm is applicable to both structured and unstructured discretizations in two and three dimensions. The method uses a tree-based hierarchical mesh with dynamic local refinement that tracks the evolving interface, concentrating resolution in a narrow band around the zero level set while preserving a coarse discretization elsewhere. The level-set field is updated through marker transport on adaptively refined meshes, enabling interface prediction and the construction of an evolving adaptive hierarchy. Concurrently, backward characteristic tracing provides accurate evaluation of the advected level-set field. Both operations employ an efficient multilevel marker and point-location algorithm to identify containing elements across refinement levels. Numerical experiments on two- and three-dimensional structured and unstructured grids, including quadrilateral, simplicial, wedge, and hexahedral meshes, subjected to severe interface deformations, demonstrate that the adaptive strategy achieves accuracy comparable to uniform discretizations at a substantially lower cost, while maintaining good conservation properties under aggressive adaptivity. Consequently, the proposed approach provides an efficient and flexible framework that can be naturally integrated into level-set-based multiphase flow solvers.
7. 从牛顿算法中学习特征：一种加速非线性参数化PDE求解器的方法
Learning features from Newton’s algorithm: a way to accelerate nonlinear parametrized PDE solvers
作者: Rémy Vallot, Florian de Vuyst, Thibault Dairay, Mathilde Mougeot
类别: cs.LG, math.AP, math.NA
发布日期: 2026-07-30
arXiv ID: 2607.28036v1
链接: http://arxiv.org/abs/2607.28036v1
摘要: It is well known that Newton’s method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.
8. 数值谱连接：通过带重采样的Koopman-Chebyshev近似识别控制偏微分方程
Numerical Spectrum Linking: Identification of Governing PDE via Koopman-Chebyshev Approximation with Resampling
作者: Phonepaserth Sisaykeo, Shogo Muramatsu
类别: math.NA, eess.SP
发布日期: 2026-07-30
arXiv ID: 2607.27728v1
链接: http://arxiv.org/abs/2607.27728v1
摘要: A numerical framework is proposed for identifying governing partial differential equations (PDEs) from observational data by establishing a link between observation-driven and equation-driven Koopman operators in a common Chebyshev spectral domain. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate Koopman operators without explicitly relating them to differential operators, the proposed framework constructs finite-dimensional Koopman operators using Chebyshev spectral representations, thereby enabling direct comparison between data-derived dynamics and candidate governing PDEs. A unified observation model together with a least-squares coefficient recovery formulation is introduced to recover Chebyshev spectral coefficients from observations obtained on arbitrary sampling grids. This provides a numerically consistent interface between practical observations and Chebyshev-based Koopman analysis. Numerical experiments under direct Chebyshev, uniform, and irregular sampling configurations demonstrate that the proposed framework accurately identifies the governing PDE from observations. An observation-density study shows that reliable PDE identification is consistently achieved once sufficient independent observations are available for stable coefficient recovery, providing a practical guideline.
9. 一种用于空间参数泊松噪声泛函的谱补偿格式：误差界与复杂度估计
A spectral-compensated scheme for space-parameter Poisson noise functionals: error bounds and complexity estimates
作者: Yun-Ching Chang
类别: math.NA, math.PR
发布日期: 2026-07-30
arXiv ID: 2607.27657v1
链接: http://arxiv.org/abs/2607.27657v1
摘要: This paper computes Poisson space noise functionals, $P^{\prime}(u)$, realised in a Gel’fand triple built from a Lévy measure $λ_β(u)du$. We isolate three discretisation parameters: a small-amplitude cut-off, a Donsker delta truncation M, and a chaos order N. For a stable-type intensity $λ_β(u)=cu^{-1-α}$ ($0<α<2$), replacing discarded small amplitudes with matched Gaussian space noise improves the Wasserstein-1 error from $O(ε^{1-α/2})$ to $O(ε)$. The residual is asymptotically normal at $O(ε^{α/2})$. This compensation reduces computational complexity from $O(τ^{-2α/(2-α)})$ to $O(τ^{-α})$. We also evaluate the Gamma-type boundary ($α=0$) and exponential tempering. Truncations converge algebraically (M) and super-geometrically (N). All predicted rates are tightly confirmed by deterministic numerical experiments via Gil-Pelaez inversion, eliminating Monte Carlo noise.
10. 一种重构$\mathbb{R}^3$中移动点声源的新型采样方法
A novel sampling method for reconstruction of a moving point acoustic source in $\mathbb{R}^3$
作者: Guanqiu Ma, Shuang Li, Hongxia Guo
类别: math.NA
发布日期: 2026-07-30
arXiv ID: 2607.27621v1
链接: http://arxiv.org/abs/2607.27621v1
摘要: In this paper, we introduce a novel sampling method to recovering the trajectory of a moving point source in R^3, where both the spatial location and emission moment of the moving point source are unknown. Combining algebraic theory with geometric knowledge, we prove the uniqueness of the source location by using measured data from five observation points. Our sampling method constructs an indicator function based on the property that the residual of the time difference of arrival constraint formula vanishes at the true source location. It achieves the reconstruction of the spatial positions and emission moments of a moving point source only using data from five observation points and their corresponding arrival times. The algorithm not only reduces the required number of observation points, but also improves computational efficiency, stability, and noise resistance. Numerical experiments verify the effectiveness of the method.
11. 可扩展的图核心集选择：基于贪心采样
Scalable Graph Coreset Selection via Greedy Sampling
作者: Zhaiming Shen, Alexander Cloninger
类别: math.NA, stat.ML
发布日期: 2026-07-30
arXiv ID: 2607.27602v1
链接: http://arxiv.org/abs/2607.27602v1
摘要: Sampling representative nodes from large graphs is fundamental to graph signal processing and network analysis, yet existing methods require access to the full graph Laplacian, making them impractical at scale. We propose a simple and effective column-selective graph sampling algorithm based on a minimum inner product greedy selection rule. At each iteration, the algorithm accesses only a small random subset of Laplacian columns, requiring no eigendecomposition or global graph traversal, making it well-suited for large-scale graphs where the full Laplacian cannot be stored in memory. We analyze the algorithm under the stochastic block model and show that, when the degree distribution is balanced across nodes, the algorithm achieves sampling proportional to cluster size, and that the resulting mean estimate is controlled for band-limited graph signals in the Paley-Wiener space, with the error decaying as inter-cluster connectivity weakens. Numerical experiments on both synthetic and real-world data validate the effectiveness of the proposed method.
12. 面向参数化线性偏微分方程的非线性流形降阶模型的残差驱动提升辨识
Residual-Driven Lifting Identification for Nonlinear-Manifold Reduced-Order Models of Parametrized Linear PDEs
作者: Francesco A. B. Silva, Jean C. Ragusa, Theron Guo, Rudy Geelen
类别: math.NA
发布日期: 2026-07-29
arXiv ID: 2607.27471v1
链接: http://arxiv.org/abs/2607.27471v1
摘要: We introduce a residual-driven procedure for training nonlinear-manifold reduced-order models for parametrized linear partial differential equations that, given prescribed latent and lifting spaces, identifies the nonlinear lifting without high-fidelity solution snapshots. The approximation is represented by a low-dimensional latent coordinate together with a nonlinear lifting into a richer reduced space. Rather than fitting the lifting to snapshot data, we determine it by minimizing a computable residual-based upper bound for the state error. For affinely parametrized operators, the resulting training objective admits an efficient offline–online decomposition, and the lifting update reduces to a sequence of low-dimensional weighted least-squares problems. Numerical evaluations on an advection–diffusion problem and a plane-strain elasticity benchmark show that the proposed approach substantially improves accuracy over linear subspaces. The resulting nonlinear models achieve accuracy comparable to snapshot-driven nonlinear-manifold training while avoiding high-fidelity snapshots in the lifting-identification stage.
13. 动态低秩平滑
Dynamical Low-Rank Smoothing
作者: Youssef Marzouk, Fabio Nobile, Fabio Zoccolan
类别: math.NA
发布日期: 2026-07-29
arXiv ID: 2607.27438v1
链接: http://arxiv.org/abs/2607.27438v1
摘要: Computational costs often make smoothing procedures prohibitive for high-dimensional data assimilation problems. To address this challenge, we propose a dynamical low-rank approximation (DLRA) methodology for smoothing concerning frameworks based on stochastic differential equations. We extend the previously developed joint mean-and-covariance optimization (JMCO) filtering setting to derive a reduced-order smoother via the Rauch–Tung–Striebel recursion and establish the corresponding Kalman–Bucy smoothing for affine drift dynamics. The resulting algorithms retain the adaptive nature of DLRA while significantly reducing the computational time and storage of the whole smoothing procedure.
14. 数据同化的动态低秩滤波器
Dynamical Low-Rank Filters for Data Assimilation
作者: Yoshihito Kazashi, Youssef Marzouk, Fabio Nobile, Fabio Zoccolan
类别: math.NA
发布日期: 2026-07-29
arXiv ID: 2607.27432v1
链接: http://arxiv.org/abs/2607.27432v1
摘要: We propose dynamical low-rank (DLR) type filters for data-assimilation problems based on stochastic differential equations (SDEs). In detail, first we derive a DLRA filter for minimizing jointly the mean and covariance error, as well as a strategy to efficiently include the relevant orthogonal directions. This last approach allows the main subspace to evolve also according to the observation operator. Those procedures naturally extend to a Kalman-Bucy type filter when dealing with linear drift, and to ensemble methods, too, resulting also suitable for problems described by nonlinear drift and possible non-Gaussian distribution. Moreover, we further propose a preliminary particle-type DLRA filter that shows potentiality in nonlinear settings. Numerical simulations show the efficacy of these procedures in relevant applications, opening up to further studies in these filtering directions.
15. 精确总变差极小化子作为守恒律高阶方法中的非振荡限制器
Exact Total Variation Minimizers as Non-Oscillatory Limiters in High-Order Methods for Conservation Laws
作者: Gabriel P. Langlois, Jerome Darbon, Rongjie Lai, Chi-Wang Shu, Xiangxiong Zhang
类别: math.NA
发布日期: 2026-07-29
arXiv ID: 2607.27416v1
链接: http://arxiv.org/abs/2607.27416v1
摘要: High-order numerical methods for conservation laws can generate spurious oscillations near discontinuities. We propose to suppress these oscillations by post-processing the numerical solution with a total variation (TV) denoising step that reduces the total variation of the nodal values. For a general analysis operator, the resulting discrete minimization problem need not satisfy the submodularity property that existing exact max-flow algorithms require, so we instead compute the TV minimizer exactly using a differential inclusion algorithm. In one dimension, the differential inclusion algorithm computes the TV minimizer in finitely many steps, requires no tuning of algorithmic parameters, and preserves the total mass. In two dimensions, we apply the 1D algorithm dimension by dimension. We test the method as a post-processing limiter for Fourier pseudospectral methods and a fifth-order finite difference scheme applied to scalar conservation laws, compressible Euler equations, and the two-dimensional incompressible Euler equations.
16. 贝叶斯反问题中的离散到连续极限
Discrete to continuum limits in Bayesian inverse problems
作者: Alexander Katsevich
类别: math.ST, math.NA
发布日期: 2026-07-29
arXiv ID: 2607.27408v1
链接: http://arxiv.org/abs/2607.27408v1
摘要: We develop a posterior-level discrete-to-continuum theory for Bayesian inverse problems whose finite-dimensional priors arise from local finite-difference regularization and whose likelihoods are of a general convex GLM-type form. In contrast to continuum-first approaches, the continuum prior and posterior are not assumed at the outset but are constructed as limits of the finite-dimensional probability measures used in numerical computation. First, with the total information parameter $τ$ and the regularization strength $κ$ fixed, we prove weak convergence on $L^2$ of the reconstructed discrete Gaussian priors and posterior measures to well-defined continuum laws. We then consider a coupled grid-refinement and small-noise limit in which $N\to\infty$ and $τ_N,κ_N\to\infty$, while $κ_N/τ_N$ remains fixed. Under explicit growth conditions relating $N$ and $τ_N$, we prove that the reconstructed discrete MAP estimates converge to the unique minimizer $u_*$ of the limiting continuum cost functional, that the reconstructed posterior measures concentrate at $u_*$, and that their centered and rescaled fluctuations converge to $\mathcal N(\vec 0,Q^{-1})$, where $Q$ is the Hessian of the continuum cost functional at $u_*$. Finally, we show that the same deterministic limit and Gaussian fluctuation law are obtained by first constructing the continuum posterior and then taking its small-noise, high-regularization limit.